We investigate the outcomes of simultaneous price competition in the presence of private information on the demand side. Each of two sellers offers a different variety of a good to a buyer endowed with a private binary signal on their relative quality. We analyze how the unique equilibrium of the game changes as a function of the (common) prior belief on the relative quality of the goods and the precision of the private information of the buyer. Competition is fierce, and the buyer enjoys high rents, when the prior belief is biased in favor of one good and private signals are not very informative: the ex ante superior seller cannot resist the temptation to clear the market, and triggers an aggressive response by the competitor. When instead the distribution of ex post valuations is highly spread, due to a vague prior belief and strong signals, the sellers become local monopolists and extract high rents from the buyer. We provide a full characterization of the mixed-strategy equilibrium which arises when the two goods are mildly differentiated ex post. Overall, the market-clearing temptation effect destroys the monotonicity and convexity of the equilibrium profit of a seller in the prior belief. As a consequence, a competing seller does not necessarily benefit from revelation of public information, sometimes even if biased in its favor. This paper analyzes the behavior of posterior distributions under the Jeffreys prior in a simultaneous equations model. The case under study is that of a general limited information setup with n + 1 endogenous variables. The Jeffreys prior is shown to give rise to a marginal posterior density which has Cauchy-like tails similar to that exhibited by the exact finite sample distribution of the corresponding LIML estimator. A stronger correspondence is established in the special case of a just-identified orthonormal canonical model, where the posterior density under the Jeffreys prior is shown to have the same functional form as the density of the finite sample distribution of the LIML estimator. The work here generalizes that of Chao and Phillips (1997), which gives analogous results for the special case of two endogenous variables.

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